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<title>Atlas software user guide -- Components</title>
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<h2>Component groups</h2>
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<i>Last updated: October 8, 2005</i>
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Let G be a connected complex reductive algebraic group, with a fixed inner
class, and fix a real form in this inner class, represented by an involution
&theta;. Let G(R) be the group of real points of G for the real form, and 
denote K the fixed point group of &theta;. As explained 
<a href="realforms.html">here</a>, K is (isomorphic to) the complexification 
of a maximal compact subgroup of G(R); in particular, the component group of
K is equal to the component group of G(R).
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<p>
The following facts are known:
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<ul>
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when G is semisimple and simply connected, the group K is connected (cf. [1],
thm. 8.1);
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<li>
if T is a maximally split &theta;-stable torus in G, then T(R) meets all 
components of G(R).
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From these facts, it follows immediately that the component group of G(R) is
always an elementary abelian 2-group (because this is true for tori), and in
fact it is possible to deduce an algorithm to compute the group from the
root datum of G and the involution &theta;. The algorithm is explained in
the 2004 notes by Fokko du Cloux <a href="http://atlas.math.umd.edu/papers">
here</a>. This information is available in the program through the
"components" command.

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<td>[1]</td>
<td>
R. Steinberg, Endomorphisms of linear algebraic groups, <i>Mem. Amer. Math.
Soc.</i> <b>80</b> (1968), pp. 1-108.
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</tr>
</table>

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<p>
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